The decimal expansion of the number $\frac{14753}{1250}$ will terminate after :

Question

Accepted Answer

To determine the number of decimal places after which the decimal expansion of the fraction $\frac{14753}{1250}$ will terminate, we need to consider the prime factorization of the denominator.

1. **Prime Factorization of the Denominator:**
   The denominator is 1250. We factorize 1250 as follows:
   $$
   1250 = 5 \times 250 = 5 \times (5 \times 50) = 5 \times (5 \times (5 \times 10)) = 5 \times (5 \times (5 \times (5 \times 2)))
   $$
   Therefore, the prime factorization of 1250 is:
   $$
   1250 = 2 \times 5^4
   $$

2. **Conditions for Terminating Decimal:**
   A fraction $\frac{p}{q}$ will have a terminating decimal expansion if and only if the denominator $q$ (in its simplest form) has no prime factors other than 2 and 5. The number of decimal places is determined by the highest power of 2 or 5 in the factorization of the denominator.

3. **Determining the Number of Decimal Places:**
   In the factorization of 1250, we have:
   - The power of 2 is 1.
   - The power of 5 is 4.

The decimal expansion will terminate after the maximum of these powers, which is 4. Therefore, the decimal expansion of $\frac{14753}{1250}$ will terminate after 4 decimal places.

4. **Explanation of Incorrect Options:**
   - **Option 1 (one decimal place):** This option is incorrect because the highest power of 2 or 5 in the factorization of 1250 is 4, not 1.
   - **Option 2 (two decimal places):** This option is incorrect because the highest power of 2 or 5 in the factorization of 1250 is 4, not 2.
   - **Option 3 (three decimal places):** This option is incorrect because the highest power of 2 or 5 in the factorization of 1250 is 4, not 3.

Thus, the correct answer is Option 4, as the decimal expansion of $\frac{14753}{1250}$ will terminate after four decimal places.